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Foundations of Computational Mathematics最新文献

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Persistence Diagram Bundles: A Multidimensional Generalization of Vineyards 持久性图束：葡萄园的多维泛化
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2026-04-28 DOI: 10.1007/s10208-026-09747-9
Abigail Hickok
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引用次数: 0
Accelerated Gradient Methods for Nonconvex Optimization: Escape Trajectories From Strict Saddle Points and Convergence to Local Minima 非凸优化的加速梯度方法：从严格鞍点的逃逸轨迹和收敛到局部极小值
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2026-04-01 DOI: 10.1007/s10208-026-09745-x
Rishabh Dixit, Mert Gürbüzbalaban, Waheed U. Bajwa
{"title":"Accelerated Gradient Methods for Nonconvex Optimization: Escape Trajectories From Strict Saddle Points and Convergence to Local Minima","authors":"Rishabh Dixit, Mert Gürbüzbalaban, Waheed U. Bajwa","doi":"10.1007/s10208-026-09745-x","DOIUrl":"https://doi.org/10.1007/s10208-026-09745-x","url":null,"abstract":"This paper considers the problem of understanding the behavior of a general class of accelerated gradient methods on smooth nonconvex functions. Motivated by some recent works that have proposed effective algorithms, based on Polyak’s heavy ball method and the Nesterov accelerated gradient method, to achieve convergence to a local minimum of nonconvex functions, this work proposes a broad class of Nesterov-type accelerated methods and puts forth a rigorous study of these methods encompassing the escape from saddle points and convergence to local minima through both an asymptotic and a non-asymptotic analysis. In the asymptotic regime, this paper answers an open question of whether Nesterov’s accelerated gradient method (NAG) with variable momentum parameter avoids strict saddle points almost surely. This work also develops two metrics of asymptotic rates of convergence and divergence, and evaluates these two metrics for several popular standard accelerated methods such as the NAG and Nesterov’s accelerated gradient with constant momentum (NCM) near strict saddle points. In the non-asymptotic regime, this work provides an analysis that leads to the “linear” exit time estimates from strict saddle neighborhoods for trajectories of these accelerated methods as well the necessary conditions for the existence of such trajectories. Finally, this work studies a sub-class of accelerated methods that can converge in convex neighborhoods of nonconvex functions with a near optimal rate to a local minimum and at the same time this sub-class offers superior saddle-escape behavior compared to that of NAG.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"64 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147586722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Crouzeix-Raviart elements on simplicial meshes in d dimensions d维简单网格上的Crouzeix-Raviart单元
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2026-02-27 DOI: 10.1007/s10208-026-09741-1
Nis-Erik Bohne, Patrick Ciarlet, Stefan Sauter
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引用次数: 0
Nearly Self-similar Blowup of Generalized Axisymmetric Navier–Stokes Equations 广义轴对称Navier-Stokes方程的近自相似爆破
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2026-02-25 DOI: 10.1007/s10208-026-09748-8
Thomas Y. Hou
{"title":"Nearly Self-similar Blowup of Generalized Axisymmetric Navier–Stokes Equations","authors":"Thomas Y. Hou","doi":"10.1007/s10208-026-09748-8","DOIUrl":"https://doi.org/10.1007/s10208-026-09748-8","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"81 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147287094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extended Regge Complex for Linearized Riemann-Cartan Geometry and Cohomology 线性化Riemann-Cartan几何与上同调的扩展Regge复形
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2026-02-17 DOI: 10.1007/s10208-026-09746-w
Snorre H. Christiansen, Kaibo Hu, Ting Lin
{"title":"Extended Regge Complex for Linearized Riemann-Cartan Geometry and Cohomology","authors":"Snorre H. Christiansen, Kaibo Hu, Ting Lin","doi":"10.1007/s10208-026-09746-w","DOIUrl":"https://doi.org/10.1007/s10208-026-09746-w","url":null,"abstract":"We show that the cohomology of the Regge complex in three dimensions is isomorphic to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ mathcal {H}^{{scriptscriptstyle bullet }}_{dR}(varOmega )otimes mathcal{R}mathcal{M}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>dR</mml:mi> </mml:mrow> <mml:mstyle> <mml:mo>∙</mml:mo> </mml:mstyle> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊗</mml:mo> <mml:mi>R</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , the de Rham cohomology of differential forms with values in infinitesimal rigid body motions. Based on an observation that the twisted de Rham complex extends the elasticity complex (based on Riemannian deformation) to the linearized version of coframes, connection 1-forms, curvature and Cartan’s torsion, we construct a discrete version of linearized Riemann-Cartan geometry on any triangulation and determine its cohomology.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"9 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146230775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The radius of statistical efficiency 统计效率的半径
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2026-02-06 DOI: 10.1007/s10208-026-09743-z
Joshua Cutler, Mateo Díaz, Dmitriy Drusvyatskiy
{"title":"The radius of statistical efficiency","authors":"Joshua Cutler, Mateo Díaz, Dmitriy Drusvyatskiy","doi":"10.1007/s10208-026-09743-z","DOIUrl":"https://doi.org/10.1007/s10208-026-09743-z","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"108 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Ulam Meets Turing: Constructing Quadratic Maps with Non-Computable Physical Measures 乌兰与图灵：构造具有不可计算物理测度的二次映射
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2026-02-02 DOI: 10.1007/s10208-026-09744-y
Cristóbal Rojas, Michael Yampolsky
{"title":"Ulam Meets Turing: Constructing Quadratic Maps with Non-Computable Physical Measures","authors":"Cristóbal Rojas, Michael Yampolsky","doi":"10.1007/s10208-026-09744-y","DOIUrl":"https://doi.org/10.1007/s10208-026-09744-y","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"253 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146101775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Spectrally Accurate Fully Discrete Schemes for Some Nonlocal and Nonlinear Integrable PDEs via Explicit Formulas 用显式公式求解非局部非线性可积偏微分方程的频谱精确全离散格式
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2026-02-02 DOI: 10.1007/s10208-026-09740-2
Yvonne Alama Bronsard, Xi Chen, Matthieu Dolbeault
{"title":"Spectrally Accurate Fully Discrete Schemes for Some Nonlocal and Nonlinear Integrable PDEs via Explicit Formulas","authors":"Yvonne Alama Bronsard, Xi Chen, Matthieu Dolbeault","doi":"10.1007/s10208-026-09740-2","DOIUrl":"https://doi.org/10.1007/s10208-026-09740-2","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146101774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Discrete Trace Theory for Non-Conforming Polytopal Hybrid Discretisation Methods 非协调多边形混合离散化方法的离散轨迹理论
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2025-12-03 DOI: 10.1007/s10208-025-09734-6
Santiago Badia, Jerome Droniou, Jai Tushar
{"title":"A Discrete Trace Theory for Non-Conforming Polytopal Hybrid Discretisation Methods","authors":"Santiago Badia, Jerome Droniou, Jai Tushar","doi":"10.1007/s10208-025-09734-6","DOIUrl":"https://doi.org/10.1007/s10208-025-09734-6","url":null,"abstract":"In this work we develop a discrete trace theory that covers non-conforming hybrid discretization methods and holds on polytopal meshes. A notion of discrete trace seminorm is defined, and <jats:italic>trace</jats:italic> and <jats:italic>lifting</jats:italic> results with respect to a discrete <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$H^1$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> -seminorm on the hybrid fully discrete space are proven. Finally, we conduct a numerical test in which we compute the proposed discrete operators and investigate their spectrum to verify the theoretical analysis. The development of this theory is motivated by the design and analysis of preconditioners for hybrid methods, e.g., of substructuring domain decomposition type.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"132 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145680181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems 热带稀疏多项式系统的零拟合性与正拟合性
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2025-12-01 DOI: 10.1007/s10208-025-09708-8
Marianne Akian, Antoine Béreau, Stéphane Gaubert
{"title":"The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems","authors":"Marianne Akian, Antoine Béreau, Stéphane Gaubert","doi":"10.1007/s10208-025-09708-8","DOIUrl":"https://doi.org/10.1007/s10208-025-09708-8","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"9 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145657532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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